That's really a very good question to ask. In short, it's because the ratio of the circumference of a circle to its radius is 2*pi. One thing to keep firmly in mind is that sine and cosine are based upon the unit circle with radius=1.
Draw out a unit circle (radius=1) centered over (0,0) on an x-y axis pair. Pick an angle, say 70 degrees, and draw a line from (0,0) out to the perimeter and stop. Call that point on the perimeter, point P. Then draw from point P straight down to the x axis so that your vertical line makes a nice 90-degree with the x axis. The height of that line is the sine of 70 degrees (or the sine of [(70/360)*2*pi] radians.)
When you ask the question, "What is the derivative of the sine(70 degrees)?" you are really asking, "What is the rate of change in the height of that line as it relates to a infinitesimally small change in degrees?" How do you try and answer that? If you look very closely at point P and think, you will see that the instantaneous _direction_ of change as you add a very tiny amount to the angle itself will be along a line that is tangent to point P. That tangent direction can be broken into two parts, one part which will directly increase the height of the line and one part that has no effect at all on it. (Those two parts are based on the sine and cosine and it is the cosine part that directly adds to the line height, by the way.)
What is the magnitude of that tangent "change?" Well, it's not hard to work out. Imagine that your angle were expressed as a value from 0 to 1 to represent 0 to 360 degrees or 0 to 2*pi radians. In other words, it's like a percentage. So let's call that % (but keep in mind it goes from 0 to 1, not 0 to 100.) Then a tiny percent change will probably produce an equivalent tiny percent change in the distance around the circumference of the circle. Put in broad strokes, if you change the number from 0 to 1, then you change the arc-length around the circumference from 0 to 2*pi.
Anyway, so how much does the sine(%) change when we add a tiny d(%) to %? Well, since the % goes from 0 to 1 while the arc length goes from 0 to 2*pi, then a tiny d(%) change in % probably causes a tiny 2*pi*d(%) change in the arc length. So, for computing the change in that line height, we'd need to write something like this:
d( sin(%) ) = cos(%) * [d(%) * 2 * pi]
or,
d( sin(%) )/d(%) = cos(%) * 2*pi
So if angles were expressed as percentages around the circle (which seems natural at first), then we'd see this 2*pi sticking out everywhere. And there would be more and more of them popping up and our equations would look a lot worse than they need to.
So to save ink and massive bouts of writer's cramp, some genius decided that maybe it might be nicer if we said, "Well, what if we expressed it as a percentage but multiplied that by 2*pi right away and just gave it the name 'radians' for kicks?" Well, then it looks easy:
d( sin( radians ) ) = cos( radians ) * d( radians )
or,
d( sin( radians ) )/d( radians) = cos( radians )
Cripes, that's going to save some ink and hand-pain.
So there it is. If you used degrees instead of percentages or radians, you'd just have yet another weird constant to deal with. So that is why degrees aren't preferred, either. It all just comes down to the sometimes annoying fact of flat world geometry that the circumference of a circle is 2*pi times its radius.